It is well known that the Mueller analysis and the Jones analysis are useful in the study of polarization optics especially relating to the state of polarization of light and polarization properties of an optical transmission medium 18. (reference [18]) Note that a list of the books and literature is referred to in this specification and is shown in Table 3 which is at the end of his specification. The advantage in using the Jones analysis is its simplicity. In spite of the fact that one can study, in principle, only perfectly polarized light in the framework of the Jones analysis, the analysis provides a simple approach compared to the other methods, e.g., Mueller's analysis. The reason for this more simplistic approach comes from the fact instead of using larger 4.times.4 or 3.times.3 matrices as in Mueller's analysis, one uses 2.times.2 matrices for the Jones analysis. The mathematical relation between the Jones' and Mueller's analysis has been studied by Takenaka in 1972 (reference [1]) and is summarized as the difference of the representation of the rotational group. When the polarization dependent loss in of a transmission media is negligible, the elements of the Mueller and Jones matrices are understood as the rotational operator. Concretely, the representations of the rotational group these matrices is known to be a three dimensional orthogonal group: O(3) and the special unitary group: SU(2). These two groups are transformed as a homomorphism mapping. The result of the comparison between the Jones analysis and the Mueller analysis is shown in Table 1. So far, in spite of the advantages explained above, most experimental study of polarization optics is achieved using the Mueller's analysis. Reasons of this dominance of the Mueller's analysis in experimental polarization optics are summarized as follows: (i) in the vector representing the light, a Stokes vector, the parameters represent optical intensities and are directly measurable quantities.(reference [1]) (ii) the Poincare sphere representation of a Stokes vector provides an intuitive way for understanding polarization. However, it is widely known that both the Jones method and Mueller's method have advantages and disadvantages relative to each other. Thus, in some instances, and in practical situations, one can approach the problem in two ways.
TABLE 1 ______________________________________ Jones' Analysis Mueller's Analysis ______________________________________ Description of the state Two dimensional Four or three dimen- of polarization complex spinors: sional real vectors: Jones vector Stokes vector Description of the optical 2 .times. 2 complex 4 .times. 4 or 3 .times. 3 real transmission media in matrices: matrices: case when the polari- Jones matrices Mueller matrices zation dependent loss is negligible Whether the state of Impossible Possible polarization is possible to determine directly or not? Theoretical Analysis Comparatively easy Complicated ______________________________________
Based on the original work by Jones (reference [3],[4]), we will briefly review the conventional matrix measurement method. For the purpose of the matrix determination experimentally, we utilize a 2.times.2 complex Jones matrix as follows: ##EQU1## From which EQU .eta..sub.4 =.beta.. (16.1) EQU .eta..sub.2 =.beta.k.sub.2 (16.2) EQU .eta..sub.3 =.beta.k.sub.4 (16.3) EQU .eta..sub.1 =.beta.k.sub.1 k.sub.4 (16.4)
Then the estimation of .eta..sub.i (i=1, 2, 3, 4) provides the 4 matrix elements. According to Jones, one measures the output electrical fields corresponding the following three incident states of polarization, represented by the Jones vectors as: ##EQU2## The output state of polarization corresponding to the above three vectors are represented as ##EQU3## Then the Jones calculus provides the following relations EQU k.sub.1 =h.sub.1 /h.sub.2 (19.1) EQU k.sub.2 =v.sub.1 /v.sub.2 (192) EQU k.sub.3 =q.sub.1 /q.sub.2 (193) EQU k.sub.4 =(k.sub.3 -k.sub.2)/(k.sub.1 -k.sub.3) (19.4)
The right hand side of these equations are represented as ratios of electric fields, and in principle are measurable quantities from which the matrix elements can be determined experimentally using the relation shown in eqs.(19) and thereafter eqs.(16). For the above discussions, the Jones matrix measurement scheme proposed by R. C. Jones in 1947 has been reviewed This measurement scheme has been utilized until now.
In the field of the optics, accurate determination of the Jones matrix makes it possible to analyze the polarization properties of some transmission media and therefore is important. In the field of the optical communications, and particularly with recent developments in optical amplifier systems, an accurate measurement of the differential group delay (hereafter referred as DGD) of the principal states of polarization (referred as PSP) is required. For this, it is necessary to know the polarization mode dispersion (PMD) characteristics of the single-mode optical fibres being utilized (references [4] and [6])
In general , it is known that the Jones matrices for an arbitrary transmission medium have the following characteristics:
(1) In case where polarization dependent loss of the medium is negligible, the Jones matrix is written in the following form (reference[6]) EQU T=exp(.rho.)U (20) PA0 (3) In case where the polarization dependent losses of the transmission medium are finite and not negligible, the determinant of the matrix U is smaller than the unity.
where P is the medium independent loss or the state of polarization of light, and U (reference[20]) can be written as: ##EQU4## where the asterisk represents the complex conjugate. (2) In the case where the transmission medium is stable, both spatially and over time, it is possible to separate a diagonal matrix which represents the linear birefringence and the effect of the circular birefringence, through a suitable unitary transformation.
From the above discussions, it can be seen that the Jones matrix representation of the medium is useful for the analysis of cases where the degree of polarization (DOP) of the transmitted lightwave is almost unity and the polarization dependent loss(PDL) of the medium are negligible. In this situation, Jones matrix elements can be described as shown in Equations 20 and 21. Based on the unimodular unitary properties of the matrix, Poole and Wagner have derived the concept of the PSP and it is now regarded that the concept plays a fundamental role in polarization mode dispersion analysis (reference[6]). Recently Aso and his colleagues have derived a theoretical formula for polarization mode dispersion measurement using a wavelength scanning method (refercnces[7]). The formula has been confirmed experimentally by using an optical fibre that consists of two polarization maintaining fibres with fmite angular misalignment. In the derivation of the formula the uniitarity characteristics of the Jones matrix have been utilized.
In relation to long haul optical communication systems having optical amplifiers and dispersion shifted fibres, or alternatively conventional single-mode fibres with dispersion compensated fibres, PMD is considered to be an important parameter which limits the transmission capacity of the system. In reference [23] the origins of polarization mode dispersion are summarized. These origins consist of intrinsic and external causes, where the external causes refer to those affects on the optical fibre which originate during cable installation. However, in practice, these causes are randomly distributed along the length of the fibre and the state of the polarization mode coupling changes with time (reference [2]). In such cases local polarization modes are coupled along the fibre and therefore the concept of the eigen states of polarization is not a viable form of analysis. This situation is called random mode coupling. Random mode coupling makes it difficult to analyze the precise nature of the polarization properties of the optical fibre. In such cases, instead of using the eigen states of polarization, the concept of the PSP is a viable form of analysis (reference[8]). Poole and Wagner have proposed a rigorous PMD evaluation method. They have proved the existence of the PSPs (principal states of polarization) both theoretically and experimentally ( reference[22]) and have shown that the DGD time of the PSPs is one of the most important parameters in PMD analysis. Based on their results, B. L. Heffnier has reported a novel measurement technique for estimating the DGD time of the PSPs (reference [4]). This method is referred to as the Jones matrix eigenanalysis (hereafter JME). According to the report from N. Gisin, JE is regarded to as one of the most precise measurement techniques (reference[24]).
In the following discussions, we will briefly review the concept of the PSP and the measurement technique of the DGD time of PSPs. Then we shall consider the problem of the proposed DGD time estimation method. As shown in references [1],[9] and [25] and references therein, the SOP changes along the optical transmission medium and is governed by the equation of the rotational motion in the spinor field (reference[20]). According to the spinorial treatment, under the assumption of perfect polarization and the negligible PDL (polarization dependent loss), the transfer matrix of the medium can be written in the form of the following 2.times.2 unitary matrix(reference[6]) ##EQU5## where the asterisk represents the complex conjugate. In this representation, we assume that the optical losses that are independent of the state of polarization are neglected. However, as shown in reference [6], it is straightforward to extend the following discussions to cases which include the influences of the optical losses and it has been confirmed that the final result of the following discussions coincide. According to the Jones calculus, the relation between the incident state of polarization (SOP) and output SOP is written as EQU .xi..sub.out =exp(i.rho.)U .xi..sub.out (23) EQU .xi..sub.in =exp(-i.rho.)U.sup.+ .xi..sub.in (24)
where + represents the henrite conjugate of the matrix and P is the absolute phase change through transmission of the medium. Taking derivatives of both sides of eq.(23) with respect to the angular frequency of light w and using the relation in eq.(24), provides the following equation ##EQU6## Here the SOP of the incident light are fixed as ##EQU7## As shown in reference [6], principal states of polarization are defined as the SOP that satisfies the following condition ##EQU8## Then the eigenvalue equation is obtained from the relation shown in eq.(25) ##EQU9## The explicit form shown in eq.(22) will lead to the following eigenvalue and corresponding eigen Jones vectors ##EQU10## The eigenvectors corresponding to these eigenvectors eigenvalues are orthogonal (reference[6]) and the vectors are called the incident PSPs (principal states of polarization). Output PSPs are obtained by operating the transfer matrix (22) on the incident PSPs, with the stability of the SOP with respect to frequency changes being first order. In order to satisfy the stability of the PSPS, the following conditions must be met: ##EQU11## where O is the Landau operator and higher than second order quantities are neglected. Principal states of polarization are stable as long as the above conditions are satisfied in practical situations. In this sense, the existence of the PSP is a generalized concept of the eigen state of polaization (reference [1]). According to reference [6], the DGD(differential group delay) time of the PSPs .DELTA.t, are one of the most important parameters describing the PMD. The above expression for the PSP allows us to obtain the following result ##EQU12## The above discussions provide a fundamental framework for PMD analysis based on the PSPs, and are provided as a theoretical background for PMD measurement using the Jones matrix eigenanalysis. One should note that the above DGD time estimation is correct when the conditions (27) are confirmed in the practical situations. In this sense, it is known that the above .DELTA.t estimation is correct to the first order approximation (reference [6]).
For the purpose of this discussion, it is necessary to describe a brief review of another estimation method for the DGD time of PSPs. This method is based on Mueller analysis and sometimes referred as the Stokes parameter method (reference [31]). According to reference [20], measurement of the frequency dependence of the output Stokes vector on the Poincare sphere will allow an estimate the DGD time of PSPs to be made. For this, use will be made of the Stokes space (reference [25]) wherein the normalized Stokes vector defines the SOP space where the geometry of the space is three-dimensional Euclidian. In the following discussions, the normalized Stokes vector is referred to simply as the Stokes vector. The concept of PSP and the DGD time of PSPs represented in the SOP field will be reviewed.
Wavelength dependence of the Stokes vector has been studied by W. Eickhoff and his colleagues (reference [26]). They have proposed the following basic equation as an analogy of the spatial SOP change in the birefringent medium by Uhicb (reference[27]). They also confirmed their own results by experiment. ##EQU13## In this equation, S is the Stokes vector of the output SOP, .omega. is the angular frequency of light. .OMEGA. a vector in the Stokes space whose magnitude coincides with the DGD time of PSPs; the two intersects of the vector with the Poincare sphere represent the PSPs. Therefore the vector is referred as the polarization dispersion vector (reference [29]). In the original paper from Eickhoff et.al.(reference[26]), the transmission media have been considered for cases where the birefingence is homogeneous along the longitudinal direction. Recently, Poole et.al., have shown that the description shown in eq.(32) can be extended straightforwardly to the case where polarization mode coupling exists. Eq.(32) is in the same form as the equation of the rotational motion(e.g., Larmor precession shown in reference[32]). Then the Stokes vector rotates on the Poincare sphere along n if the polarization dispersion vector is independent of the wavelength.
As discussed previously, the original Jones matrix estimation method developed by Jones is very sensitive to the unavoidable experimental errors that violate the symmetry properties of the matrix. Then the unitarity of the estimated matrix is generally violated. Principally the reason of the symmetry breaking could be considered as follows:(a) According to the original method, matrix elements are determined individually; then the unavoidable measurement error directly influences the results, (b)When the PDL(polarization dependent losses) of the medium are not perfectly neglected, the effect of PDL violates the symmetry of the matrix, (e) Degradation of the DOP (degree of polarization) of the output light makes it difficult to maintain consistency between experiments and theory. The matrix symmetry breakdown causes difficulties in obtaining certain important information related to the birefringence of the medium.
As discussed previously, Poole and Wagner have derived the concept of PSP (principal state of polarization ) based on the Jones analysis. According to their results, B. L. Heffner developed the PMD estimation technique (reference[4]). According to Heffhier, he has estimated the Jones matrix using the same approach as the original matrix estimation method (reference[3]). Thus, he could not avoid the matrix symmetry breaking down. However, in order to avoid the difficulties, Heffner has introduced another eigenvalue equation based on a exponential approximation (reference [4]). However since the original PSP theory does not demand his exponential approximation, the following problem has remained: Is the exponential approximation consistent with the first order approximation of the PSP or not? Aso and his colleagues have investigated the difference of the first order approximation region and the exponential approximation region in the wavelength domain. Results of their study using an 80km length dispersion shifted single-mode optical fibre showed that any differences between the sufficient condition of these two approximations could not be found in the frequency domain measurements (reference [10],[11]). However their discussions were never extended straightforwardly to the general case. Thus the importance of maintaining the Jones matrix symmetry is noted in this specification.
As described heretofore, B. L. Heffner has published a design of the PMD measurement system based on the concept of the principal state of polarization (reference [4]). The method is referred as the Jones matrix eigenanalysis. A summary is now provided of the measurement scheme proposed by Hefffier. Firstly, three different SOPs (States of Polarization) with angular frequency .omega..sub.o are launched into a transmission medium under test. Then information of the corresponding output SOP leads us to determine every Jones matrix component of the medium uniquely according to the scheme shown in the previous discussions (reference [3]). According to Heffner, one can construct a characteristic finite differential equation corresponding to the eigenvalue equation (28) from the information of the Jones matrix measured at two frequencies with finite interval .DELTA..omega., say at .omega..sub.o and .omega..sub.o +.DELTA..omega.. The solution of the finite differential equation allows the DGD (differential group delay) time of PSPs to be estimated. Within the framework of the first order approximation criteria, frequency step-size .DELTA..omega. for the measurement is necessary in order to ensure the existence of the PSPs as the base vector in the Stokes space. However, in practice the measurement accuracy demands a large .DELTA..omega.. Nevertheless one can not distinguish the difference of the information between the measurements taken at two frequencies if the stepsize is too small. The above discussion indicates the existence of a suitable frequency stepsize for the PMD measurement from Jones matrix eigenanalysis, wherein a suitable frequency step-size is the largest step-size in which the PSP remains in the first order(reference[30]).
Incorrectly selecting the step-size apparently causes a DGD time mis-estimation. This phenomenon is noted in Hewlett Packard's polarization analyzer user's manual (reference[30]) and is also confirmed in experiments by Aso, etal. (reference[10],[11]). This drawback is considered as the most conspicuous in the Jones matrix eigenanalysis. A technique which has been developed in order to solve the above problem will now be discussed (reference [4],[5],[31]). The first step of this technique is to observe how the trajectory of the output SOP (state of polarization) changes with respect to the angular frequency of light. Based on the equation (32) and (33), the first order frequency change .DELTA..omega. leads us to observe tie trajectory where the Stokes vector traces the arc of a complete circle as shown in FIG. 5. If the choice of the frequency step-size .DELTA..omega. is larger than the "suitable" stepsize, the trajectory runs off the arc of the above reference circle. The suitable stepsize can thus be estimated as the largest frequency step-size where the observed Stokes vector trajectory is on the circle (reference [10]). However, the exact proof that the estimated DGD (differential group delay) time of PSPs (principal states of polarization) using the above .DELTA..omega. by the Jones matrix eigenanalysis seems to have not been given.
According to the Jones matrix eigenanalysis approach reported in reference [4]and[5], the transmission matrix is determined by the original method proposed by R. C. Jones (reference[3]). However, the negligibly small PDL (polarization dependent losses) of the medium and the unavoidable measurement errors may violate the unitarity of the estimated matrix.
Heffner has derived the eigenvalue equation that differs from equation (28); in his equation two eigenvalues coincide with the DGD time of PSPs. His equation is ingenious and the constraint of the matrix symmetry does not explicitly appear. Furthermore he has introduced exponential approximation which did not appear in the original paper (reference[6]). The exponential approximation leads to a convenient analysis. However the scheme seems to have no relation with the first order approximation. Aso and his colleagues have studied the difference of the exponential approximation with first order approximation in the PSP theory. They have measured the difference by using an 80 km dispersion shifted single mode optical fibre (reference [10],[11]). In these experiments, they found no difference between the two approximations. However, their experiments employed a transmission medium and no general proof was provided.
In order to determine a suitable wavelength step-size for PMD (polarization mode dispersion) measurement by Jones matrix eigenanalysis, Aso and his colleagues have proposed the comparison of global and local measurements (reference[11]). However, when one applies the idea to practical measurements, measurements over long durations of time are necessary to determine a suitable .DELTA..omega. and the constraints from the local measurements limit the measurable DGD time of PSPs to less than 5 ps.
What follows is a summary of the causes of the drawbacks of the JME (Jones Matrix Eigenanalysis) method. (a) Measurement apparatus not made precisely according to the original PSP theory obtained by Poole and Wagner (reference [6]).(b) Since the original PSP theory is expressed within the framework of the Jones' calculus, in order to realize the measurements based on the theory, it is necessary to represent the explicit relation between the Jones vector and measurable quantities(e.g., Stokes parameters). Frequency dependence of the output SOP described by the Stokes vector is expressed by the eq.(32). The left hand side of eq.(32): ds/d.omega. corresponds to the derivative of the Jones vector with respect to the angular frequency. However to date the relationship between the eq.(32) and the corresponding equation expressed in the framework of the Jones' calculus seems not to have been analyzed precisely. (c) Also the relationship between the polarization dispersion vector .OMEGA. and the Jones matrix U has not yet been studied.
According to the Jones' calculus, the Jones vector representation of the output principal states is obtained by operating the Jones matrix on the incident Jones vector. The matrix is of 2.times.2 form. On the other hand, according to the Mueller's calculus, the output Stokes vector is obtained in a similar manner as the Jones calculus. However in this case the matrix is 4.times.4 and the number of matrix elements increase. Thus, the Jones calculus makes it easy to realize the calculation and obtain the output SOP. Jones vector measurements techniques which are consistent with the Stokes vector measurements have not yet been demonstrated.